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Compressed Sensing (2016)

pyrunner.com

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Re: Compressed Sensing (2016)

#11

Post by flaviu2 »

notajoke wrote: post #5

How does the audio example square with nyquist limit? Basically you can get it back, mostly, with some clever tricks, but I’m not crazy thinking that the original was decimated beyond lossless recovery right?

> Compressed sensing in this context is made possible by the fact that the signal’s frequency content is highly sparse.

The fact that the signal is so regular is what makes this possible. You're sampling the same signal many times, far more than twice the frequency of the highest frequency.

If the signal's frequencies changed over time or if there were more frequencies in the signal, this wouldn't work or would require far more data.

Re: Compressed Sensing (2016)

#13

Post by SimplyUnknown »

This is a Big Deal in MRI image reconstruction. As the MR scanner samples in the frequency domain (or k-space, in the nomenclature) one can significantly accelarete MR exams by using C/S reconstruction. The only restriction is that the sampling mask is random. This causes incoherent artifacts in image space, which can be removed by denoising the image in a sparse domain, e.g. wavelets. See also the work by Lustig et al. https://onlinelibrary.wiley.com/doi/full/10.1002/mrm.21391

Re: Compressed Sensing (2016)

#14

Post by im3w1l-alt »

notajoke wrote: post #5

How does the audio example square with nyquist limit? Basically you can get it back, mostly, with some clever tricks, but I’m not crazy thinking that the original was decimated beyond lossless recovery right?

The nyquist limit is based on recovering a general signal. If you have a-priori knowledge about the signal you can do much better.

Re: Compressed Sensing (2016)

#15

Post by timeinput »

hatsunearu wrote: post #8
timeinput wrote: post #7

Earlier quoted context omitted.

I thing DCT is a special case (only real) version of a Fourier transform.

Well, kind of. The FT on a real signal gives you a symmetric spectrum that is complex. IDK seems kinda weird, there's a lot of handwavey stuff that I don't fully understand. And I'm also wondering what the significance of the frequency domain is--can you generalize compressed sensing with other transforms as well?

Definitely. JPEG2000 uses a wavelet transform instead of the DCT to achieve similar goals.

One way to think of it is your transforming to some other basis. You could do something similar with any vector base change where you go from a dense representation to a sparse one. Think of things like an eigen decomposition where most of the eigenvalues are small or zero and can be ignored

Re: Compressed Sensing (2016)

#16

Post by hatsunearu »

timeinput wrote: post #15
hatsunearu wrote: post #8

Earlier quoted context omitted.

Well, kind of. The FT on a real signal gives you a symmetric spectrum that is complex. IDK seems kinda weird, there's a lot of handwavey stuff that I don't fully understand. And I'm also wondering what the significance of the frequency domain is--can you generalize compressed sensing with other transforms as well?

Definitely. JPEG2000 uses a wavelet transform instead of the DCT to achieve similar goals. One way to think of it is your transforming to some other basis. You could do something similar with any vector base change where you go from a dense representation to a sparse one. Think of things like an eigen decomposition where most of the eigenvalues are small or zero and can be ignored

Yeah. do they like make up tailored wavelet transforms just to get it nice and sparse?

Or does it have to be sparse? I thought it just has to have a low L1 norm

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